Our approach leverages the idea of state augmentation that incorporates the quantile value as an
extra state variable, which has been suggested and utilized in prior work [1]}, [2]}, [3]}, [4]}, but in exclusively discrete-time settings. To our knowledge, our work is
the first to apply this in continuous-time, which introduces new technical challenges. We clearly
take advantage of the continuous-time setting: the martingale representation theorem, which is
intrinsically continuous-time, is fundamental in our analysis as it allows us to parameterize the
adversary's control policy with a real-valued stochastic process that can be tractably optimized.
Moreover, the choice of continuous-time is critical in this application, since the optimization of
an analogous discrete-time model would involve a Bellman equation that is not separable, and hence
is unlikely to admit an analytic solution (see the discussion at the end of §REF ).
